Решение №17047: \(\left ( \frac{\sqrt[3]{mn^{2}}+\sqrt[3]{m^{2}n}}{\sqrt[3]{m^{2}}+2\sqrt[3]{mn}+\sqrt[3]{n^{2}}}-2\sqrt[3]{n}+\frac{m-n}{\sqrt[3]{m^{2}}-\sqrt[3]{n^{2}}} \right ):\left ( \sqrt[6]{m}+\sqrt[6]{n} \right )=\frac{\sqrt[3]{m}-\sqrt[3]{n}}{\sqrt[6]{m}+\sqrt[6]{n}}=\frac{\left ( \sqrt[6]{m}+\sqrt[6]{n} \right )\left ( \sqrt[6]{m}-\sqrt[6]{n} \right )}{\sqrt[6]{m}+\sqrt[6]{n}}=\sqrt[6]{m}-\sqrt[6]{n}\)

Ответ: \(\sqrt[6]{m}-\sqrt[6]{n}\)

Решение №17048: \(\frac{\left ( \sqrt{p^{3}}:\sqrt{p}+p \right )^{\frac{1}{4}}:\sqrt[8]{\left ( p-q \right )^{3}}}{\left ( \frac{\sqrt{q}}{\sqrt{p}-\sqrt{q}}-\sqrt{\frac{q}{p}}+1 \right )^{\frac{1}{4}}}=\frac{\sqrt[4]{\frac{\sqrt{p^{3}}}{\sqrt{p}}+p}:\sqrt[8]{\left ( p-q \right )^{3}}}{\sqrt[4]{\frac{\sqrt{q}}{\sqrt{p}-\sqrt{q}}-\frac{\sqrt{q}}{\sqrt{p}}+1}}=\sqrt[8]{\frac{\left ( \left ( \sqrt{p}+\sqrt{q} \right )\left ( p-\sqrt{pq}+q \right ) \right )^{2}\left ( \sqrt{p}-\sqrt{q} \right )^{2}}{\left ( p-q \right )^{3}\left ( p-\sqrt{pq}+q \right )^{2}}}=\sqrt[8]{\frac{\left ( \sqrt{p}+\sqrt{q} \right )^{2}\left ( \sqrt{p}-\sqrt{q} \right )^{2}}{\left ( p-q \right )^{3}}}=\sqrt[8]{\frac{\left ( p-q \right )^{2}}{\left ( p-q \right )^{3}}}=\sqrt[8]{\frac{1}{p-q}}-\frac{1}{\sqrt[8]{p-q}}\)

Ответ: \(\sqrt[8]{\frac{1}{p-q}}-\frac{1}{\sqrt[8]{p-q}}\)

Решение №17049: \(\sqrt[n]{y^{\frac{2n}{m-n}}}:\sqrt[m]{y^{\frac{\left ( m-n \right )^{2}+4mn}{m^{2}-n^{2}}}}=y^{\frac{2n}{n\left ( m-n \right )}}:y^{\frac{m^{2}-2mn+n^{2}+4mn}{m\left ( m+n \right )\left ( m-n \right )}}=y^{\frac{2}{m-n}}:y^{\frac{\left ( m+n \right )^{2}}{m\left ( m+n \right )\left ( m-n \right )}}=y^{\frac{2}{m-n}-\frac{m+n}{m\left ( m-n \right )}}=y^{\frac{2m-m-n}{m\left ( m-n \right )}}=y^{\frac{1}{m}}=\sqrt[m]{y}\)

Ответ: \(\sqrt[m]{y}\)

Решение №17050: \(\frac{x^{\frac{3}{p}}-x^{\frac{3}{q}}}{\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )^{2}-2x^{\frac{1}{q}}\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )}+\frac{x^{\frac{2}{p}}}{x^{\frac{q-p}{pq}}+1}=\frac{\left ( x^{\frac{1}{p}}-x^{\frac{1}{q}} \right )\left ( x^{\frac{2}{p}}+x^{\frac{1}{p}}x^{\frac{1}{q}}+x^{\frac{2}{q}} \right )}{\left ( x^{\frac{1}{p}}+x^{\frac{1}{q}} \right )\left ( x^{\frac{1}{p}}-x^{\frac{1}{q}} \right )}+\frac{x^{\frac{1}{p}}}{\frac{x^{\frac{1}{p}}}{x^{\frac{1}{q}}}+1}=x^{\frac{1}{p}}+x^{\frac{1}{q}}=\sqrt[p]{x}+\sqrt[q]{x}\)

Ответ: \(\sqrt[p]{x}+\sqrt[q]{x}\)

Решение №17051: \(\frac{2a\left ( a+2b+\sqrt{a^{2}+4ab} \right )}{\left ( a+\sqrt{a^{2}+4ab} \right )\left ( a+4b+\sqrt{a^{2}+4ab} \right )}=\frac{a+\sqrt{a^{2}+4ab}}{a+4b+\sqrt{a^{2}+4ab}}=\frac{a^{2}+4ab-a\sqrt{a^{2}+4ab}+a\sqrt{a^{2}+4ab}+4b\sqrt{a^{2}+4ab}-a^{2}-4ab}{\left ( a+4b \right )^{2}-\left ( \sqrt{a^{2}}+4ab \right )^{2}}=\frac{4b\sqrt{a^{2}+4ab}}{a^{2}+8ab+16b^{2}-a^{2}+4ab}=\frac{4b\sqrt{a^{2}+4ab}}{4ab+16b^{2}}=\frac{4b\sqrt{a^{2}+4ab}}{4b\left ( a+4b \right )}=\frac{\sqrt{a\left ( a+4b \right )}}{a+4b}=\sqrt{\frac{a\left ( a+4b \right )}{\left ( a+4b \right )^{2}}}=\sqrt{\frac{a}{a+4b}}\)

Ответ: \(\sqrt{\frac{a}{a+4b}}\)

Решение №17052: \(\frac{t^{2}-t-6-\left ( t+3 \right )\sqrt{t^{2}-4}}{t^{2}+t-6-\left ( t-3 \right )\sqrt{t^{2}-4}}=\frac{\left ( t^{2}-4 \right )-\left ( t+2 \right )-\left ( t+3 \right )\sqrt{t^{2}-4}}{\left ( t^{2}-4 \right )+\left ( t-2 \right )-\left ( t-3 \right )\sqrt{t^{2}-4}}=\frac{\left ( t-3 \right )\left ( t+2 \right )-\left ( t+3 \right )\sqrt{\left ( t-2 \right )\left ( t+2 \right )}}{\left ( t+3 \right )\left ( t-2 \right )-\left ( t-3 \right )\sqrt{\left ( t-2 \right )\left ( t+2 \right )}}=-\frac{\sqrt{t+2}}{\sqrt{t-2}}=-\sqrt{\frac{t+2}{t-2}} \)

Ответ: \(-\sqrt{\frac{t+2}{t-2}}\)

Решение №17053: \(\frac{x^{2}+2x-3+\left ( x+1 \right )\sqrt{x^{2}-9}}{x^{2}-2x-3+\left ( x-1 \right )\sqrt{x^{2}-9}}=\frac{\left ( x^{2}-1 \right )+2\left ( x-1 \right )+\left ( x+1 \right )\sqrt{x^{2}-9}}{\left ( x^{2}-1 \right )-2\left ( x+1 \right )+\left ( x-1 \right )\sqrt{x^{2}-9}}=\frac{\left ( x+3 \right )\left ( x-1 \right )+\left ( x+1 \right )\sqrt{\left ( x-3 \right )\left ( x+3 \right )}}{\left ( x-3 \right )\left ( x+1 \right )+\left ( x-1 \right )\sqrt{\left ( x-3 \right )\left ( x+3 \right )}}=\frac{\sqrt{x+3}}{\sqrt{x-3}}=\sqrt{\frac{x+3}{x-3}}\)

Ответ: \(\sqrt{\frac{x+3}{x-3}}\)

Решение №17054: \(\frac{\left ( \left ( x+2 \right )^{-\frac{1}{2}} +\left ( x-2 \right )^{-\frac{1}{2}}\right )^{-1}+\left ( \left ( x+2 \right )^{-\frac{1}{2}}-\left ( x*2 \right )^{-\frac{1}{2}} \right )^{-1}}{\left ( \left ( x+2 \right )^{-\frac{1}{2}}+\left ( x-2 \right )^{-\frac{1}{2}} \right )^{-1}-\left ( \left ( x+2 \right )^{-\frac{1}{2}}-\left ( x-2 \right )^{-\frac{1}{2}} \right )^{-1}}=\frac{\left ( \frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{x-2}} \right )^{-1}+ \left ( \frac{1}{\sqrt{x+2}}-\frac{1}{\sqrt{x-2}} \right )^{-1}}{ \left ( \frac{1}{\sqrt{x+2}}+\frac{1}{\sqrt{x-2}} \right )^{-1} -\left ( \frac{1}{\sqrt{x+2}}-\frac{1}{\sqrt{x-2}} \right )^{-1}}=\frac{\frac{\sqrt{x-2}-\sqrt{x+2}+\sqrt{x-2}+\sqrt{x+2}}{\left ( \sqrt{x-2}+\sqrt{x+2} \right )\left ( \sqrt{x-2}-\sqrt{x+2} \right )}}{\frac{\sqrt{x-2}+\sqrt{x+2}-\sqrt{x-2}+\sqrt{x+2}}{\left ( \sqrt{x-2}+\sqrt{x+2} \right )\left ( \sqrt{x-2}-\sqrt{x+2} \right )}}=-\frac{\sqrt{x-2}}{\sqrt{x+2}}=-\sqrt{\frac{x-2}{x+2}}\)

Ответ: \(-\sqrt{\frac{x-2}{x+2}}\)

Решение №17055: \(\sqrt{\frac{p^{2}-q\sqrt{p}}{\sqrt{p}-\sqrt[3]{q}}+p\sqrt[3]{q}}\left ( p+\sqrt[6]{p^{3}q^{2}} \right )^{-\frac{1}{2}}=\sqrt{\sqrt{p}\left ( \sqrt{p^{2}}+\sqrt{p}\sqrt[3]{q}+\sqrt[3]{q^{2}}\right )+p\sqrt[3]{q}}\cdot \frac{1}{\sqrt{\sqrt{p}\left ( \sqrt{p}+\sqrt[3]{q} \right )}}=\sqrt{\frac{\left ( \sqrt{p}+\sqrt[3]{q} \right )^{2}}{\sqrt{p}+\sqrt[3]{q}}}=\sqrt{\sqrt{p}+\sqrt[3]{q}}\)

Ответ: \(\sqrt{\sqrt{p}+\sqrt[3]{q}}\)

Решение №17056: \(\left ( \sqrt{1-x^{2}}+1 \right ):\left ( \frac{1}{\sqrt{1+x}}+\sqrt{1-x} \right )=\left ( \sqrt{1-x^{2}}+1 \right ):\frac{1+\sqrt{1-x^{2}}}{\sqrt{1+x}}=\frac{\left ( \sqrt{1-x^{2}}+1 \right )\sqrt{1+x}}{1+\sqrt{1-x^{2}}}=\sqrt{1+x}\)

Ответ: \(\sqrt{1+x}\)